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18 II REAL NUMBERS

2. ORDER PROPERTIES OF THE REAL NUMBERS

The relation jc < y is equivalent to "x< y or x = y"

(2.2.1) For any pair of real numbers x, y, one and only one of the three relations
x <y, x =y, x>y holds.

This follows from (II.3) and (II.2), for if x ^ y, it is impossible that
x < y and x> y hold simultaneously by (II.2).

(2.2.2) The relations "xi£y and y < z" and "x<y and ;><z" both
imply x <z.

For by (II. 1) they imply x ^ z, and if we had x = z, then we would
have both x^y and y < x (or both x <y and y ^ x) which is absurd.

(2.2.3) Any finite subset A 0/ R has a greatest element b and a smallest
element a (i.e., a ^ x < b for every x € A).

We use induction on the number n of elements of A, the property being
obvious for n = 1. Let c be an element of A, B = A — {c}; B has n — 1
elements, hence a smallest element a' and a greatest element b'. If a¹ ^ c ^ b\
a' is the smallest and b' the greatest element of A; if b' < c, c is the greatest
and a' the smallest element of A; if c ^ a', c is the smallest and b' the greatest
element of A.

(2.2.4) If A is a finite subset o/R having n elements, there is a unique bijection
f of the set l_n of integers i such that l^i^n, onto A, such that /(/) <f(j)
for i <j (fis called the natural ordering of A).

Use induction on n, the result being obvious for n = 1. Let b be the
greatest element of A (2.2.3), and B = A - {b}; let g be the natural ordering
of B. Any mapping/of I_n onto A having the properties stated above must
be such that/(«) = 6, and therefore /(I^-O = B; hence/must coincide on
!„_! with the natural ordering g of B, which shows/is unique; conversely,
defining/as equal to g in l_n.₁ and such that/(/i) = b, we see at once that/
has the required properties.



	18 II REAL NUMBERS 2. ORDER PROPERTIES OF THE REAL NUMBERS The relation jc < y is equivalent to "x< y or x = y"

	(2.2.1) For any pair of real numbers x, y, one and only one of the three relations x <y, x =y, x>y holds. This follows from (II.3) and (II.2), for if x ^ y, it is impossible that x < y and x> y hold simultaneously by (II.2).

	(2.2.2) The relations "xi£y and y < z" and "x<y and ;><z" both imply x <z. For by (II. 1) they imply x ^ z, and if we had x = z, then we would have both x^y and y < x (or both x <y and y ^ x) which is absurd.

	(2.2.3) Any finite subset A 0/ R has a greatest element b and a smallest element a (i.e., a ^ x < b for every x € A). We use induction on the number n of elements of A, the property being obvious for n = 1. Let c be an element of A, B = A — {c}; B has n — 1 elements, hence a smallest element a' and a greatest element b'. If a¹ ^ c ^ b\ a' is the smallest and b' the greatest element of A; if b' < c, c is the greatest and a' the smallest element of A; if c ^ a', c is the smallest and b' the greatest element of A.

	(2.2.4) If A is a finite subset o/R having n elements, there is a unique bijection f of the set l_n of integers i such that l^i^n, onto A, such that /(/) <f(j) for i <j (fis called the natural ordering of A). Use induction on n, the result being obvious for n = 1. Let b be the greatest element of A (2.2.3), and B = A - {b}; let g be the natural ordering of B. Any mapping/of I_n onto A having the properties stated above must be such that/(«) = 6, and therefore /(I^-O = B; hence/must coincide on !„_! with the natural ordering g of B, which shows/is unique; conversely, defining/as equal to g in l_n.₁ and such that/(/i) = b, we see at once that/ has the required properties.